There are a few options, depending on how much automation you are looking for.
I've played and run Lancer using a white board app and Comp Con and that worked great. I'm currently running Lancer using Foundry VTT which also works great.
The advantage to a white board style setup is that it's very portable (maps are just images with no inherent game logic) and you can use whatever application you and your friends enjoy, from Playingcards.io to Miro to Excalidraw to Google Slides. White board apps usually don't require special accounts for anyone except the host and some (like Miro and Excalidraw) support a massive, scalable canvas that can fit as many notes, maps, drawings, and references images as you'd like to throw at it. Excalidraw has the additional advantage of letting you download the current room and re-upload it later, so you can even set up multiple different boards ahead of time even on a free account without the ability to save them to the cloud.
Comp/Con (https://compcon.app/#/) does everything else. It tracks character builds, tracks combat encounters, and can even track a whole mission sequence if you like. The interface is slick and has lots of tooltips and reference materials to make understanding new abilities and interactions really easy.
Foundry (https://foundryvtt.com/) is a fully-feature VTT with support for lots of different games both through official and fan-made modules. For Lancer, there is an excellent community maintained module as well as a host of addons that make it especially fun to use. As a fully featured VTT, setting up maps is a bit more work--you need to get the map background scaled correctly to tokens and lined up properly with the hex grid to make the most out of the automations Foundry provides, but you *can* also work without a software-enforced grid and simply count hexes by hand on a map with visual hexes to save yourself the time at the expense of missing out on some useful features. The foundry module has integration with Comp/Con to import content you've downloaded or purchased in the form of LCPs and import pilots you've made on Comp/Con.
I'm currently running Operation Solstice Rain and I took the time to set the terrain heights properly on all of the maps so I can make the most of a robust line of sight tool. I also used community modules that required minimal setup to add animations and sound effects to most tech and weapon attacks as well as automatically replace destroyed mechs with rubble after playing a big explosion effect. All thanks to the hard work of the Lancer community! It takes more time to get things set up than with Comp/Con, but once you get used to some of Foundry's quirks, it has a lot to offer both specifically for Lancer and for RPGs in general.
Foundry costs $50 for the GM, after which you can host games by direct IP connection. Players connect through a web interface similar to that used by other web-based VTTs like Roll20, Role, and so forth. Players can connect as long as the client is running on your computer and your firewall /router allows traffic on the port that you set. I go one step further, running the server on a Raspberry Pi and hosting it through a reverse-proxy on my own domain/website. It gives me a bit more control over how and when the server is running--players can connect to the server even when I'm not online to make notes or update their characters and I can run the game from anywhere on any computer, without needing to have the client installed on the computer I'm using to run the game. Foundry does not currently support mobile devices, as far as I'm aware.